l-adic Cohomology of the Dual Lubin-Tate Tower via the Exterior Power

Exterior Power

Thesis by

Foo Yee Yeo

In Partial Fulfillment of the Requirements for the

degree of

Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY

Pasadena, California

2016

© 2016

Foo Yee Yeo

ACKNOWLEDGEMENTS

I would like to thank my advisor, Elena Mantovan, for suggesting this problem, and

for her invaluable help, guidance and support throughout my work on this problem,

without which this thesis would not have been possible. I would also like to thank her

for carefully reading through various drafts of this thesis, and suggesting numerous improvements and corrections. I would also like to thank Hadi Hedayatzadeh for his

ABSTRACT

In this thesis, we study thel-adic cohomology of the dual Lubin-Tate tower by using the exterior power of a πL-divisible OL-module to relate it to the cohomology of

the Lubin-Tate tower. By using a result of Harris-Taylor on the cohomology of

the Lubin-Tate tower, we show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower realizes the local Langlands and the Jacquet-Langlands

TABLE OF CONTENTS

Acknowledgements . . . iii

Abstract . . . iv

Table of Contents . . . v

Chapter I: Introduction . . . 1

Chapter II: Background onπ_L-divisibleO_L-modules . . . 9

2.1 Ramified Witt vectors, Dieudonné O_L-modules andO_L-displays . . . 9

2.2 IsoclinicπL-divisibleOL-modules and their endomorphism algebras 10 2.3 Serre dual and exterior powers ofπ_L-divisibleO_L-modules . . . 12

Chapter III: Moduli spaces ofπL-divisibleOL-modules . . . 15

3.1 Basic Rapoport-Zink spaces . . . 15

3.2 Results on the Lubin-Tate tower . . . 17

Chapter IV: The endomorphism algebras,GLhand representation theory . . . 19

4.1 The endomorphism algebras andGLh . . . 19

4.2 Representation theory . . . 27

Chapter V: The duality map and the exterior power map on moduli spaces . . 31

5.1 Definitions of the duality map and the exterior power map . . . 31

5.2 Properties of the exterior power map and group actions . . . 32

Chapter VI: The geometry and cohomology of the dual Lubin-Tate tower . . . 35

6.1 Geometry of the dual Lubin-Tate tower . . . 35

6.2 Cohomology of the dual Lubin-Tate tower . . . 38

C h a p t e r 1

INTRODUCTION

Generalizing earlier work of Lubin-Tate ([LT66]) and Drinfeld ([Dri74]),

Rapoport-Zink ([RZ96]) constructed moduli spaces of deformations ofp-divisible groups for data of type EL and PEL. These spaces, which are now known as Rapoport-Zink

spaces, are local models of Shimura varieties. Kottwitz (cf. [Rap95]) conjectured

that thel-adic cohomology of basic Rapoport-Zink spaces should partly realize the local Langlands and the Jacquet-Langlands correspondences.

A particularly well-known example of a Rapoport-Zink space is the Lubin-Tate

tower. Let p be an odd prime, L be a finite extension of _Qp with uniformizer πL

and residue field kL = Fq, and letG1,h−1be the isoclinic πL-divisible OL-module

over Fp of dimension 1 and height h ≥ 2. Rapoport-Zink ([RZ96]) showed that,

in this special case, the functor of deformations of G1,h−1 with a quasi-isogeny is representable by a formal scheme LTh, which is non-canonically isomorphic to

`

i∈ZSpfOL˘[[t1, . . . ,th−1]], whereO_L˘ is the ring of integers of ˘L = L·_QLnr_p. We get

a tower of étale covers over the rigid analytic generic fiber ofLThby trivializing the

Tate module of the universalπ_L-divisibleO_L-module. LetLT_hnbe the étale cover that trivializes the Tate module of the universalπL-divisibleO_L-module mod Γn, where

Γnis the subgroup ofGLh(OL)consisting of matrices congruent toIhmod πn_L. The

projective systemLT_h∞ = {LT_hn}_nis known as the Lubin-Tate tower.

Let C be the completion of an algebraic closure of ˘L. After base change to C, the Lubin-Tate tower has actions by the three groups GLh(L), D× (where D =

End(G1,h−1) ⊗Zp Qp is the central division algebra over L of invariant 1/h) and

WL (the Weil group of L). By using global methods involving Shimura varieties,

Harris-Taylor ([HT01]) showed that the supercuspidal part of the cohomology of

the Lubin-Tate tower realizes the local Langlands correspondence for GLh and

the Jacquet-Langlands correspondence for D×, up to some twists, thus verifying Kottwitz’s conjecture in this particular case.

If, in the above moduli problem, we replace G1,h−1 by Gh−1,1, the isoclinic πL

by the groupsGLh(L), E×(whereE = End(Gh−1,1)⊗ZpQp) andWL.

The aim of this paper is to understand the cohomology of the dual Lubin-Tate tower

M_h∞by relating it to the cohomology of the Lubin-Tate towerLT_h∞. We will show that the cohomology of the dual Lubin-Tate tower also realizes the Jacquet-Langlands

and the local Langlands correspondences, up to certain twists.

For the rest of the introduction, we will slightly abuse notation, and not distinguish

between the Lubin-Tate towerLT_h∞, which is defined over ˘L, and its base change to C. We will also do the same for the dual Lubin-Tate tower M_h∞. Although some of the statements still hold even without base change, for simplicity, the reader may

assume (for the rest of the introduction) that we are always working with the base

change toC.

As usual, letlbe an odd prime different fromp. We consider the functor

H_ci(M_h∞): RepE× → Rep(GLh(L)×WL) ρ7→ HomE×(Hi

c(M

∞

h ,Ql), ρ).

The following are the main results:

Theorem A. Forπ ∈Cusp(GLh(L)),

H_ch−1(M_h∞)cusp(JL−1(π)) =π ⊗rec(π∨⊗ (χπ◦det)⊗ (| · | ◦det)

h−1

2 )(h−1),

where χπ is the central character of π, H_ch−1(M_h∞)cusp is the supercuspidal part

of H_ch−1(M_h∞), rec is the local Langlands correspondence and JL is the Jacquet-Langlands correspondence.

Theorem A is a consequence of the following Theorem which expresses the

co-homology of the dual Lubin-Tate tower M_h∞ in terms of the cohomology of the Lubin-Tate towerLT_h∞. For convenience, we will fix certain embeddings of the divi-sion algebrasD×andE×intoGLh(Lh)(see Section 4.1), whereLhis the unramified

extension of degreehoverL.

Theorem B. Define

θ :D× → E× ψ :GLh(L)→ GLh(L)

Let θ∗H_ci(LT_h∞,Ql) be the pushforward of the representation Hci(LT

∞

h ,Ql) under the map

θ :D× θ(D×).

Then for alli ≥ 0,

ψ∗

ResGLh(L) ψ(GLh(L))H

i c(M

∞

h,Ql)

IndE × θ(D×₎θ∗*

,

H_ci(LT_h∞,Ql)

K +

-asE××GLh(L)×WL representations, whereK =ker(θ).

To prove Theorem B, we first define a map fromLT_h∞to M_h∞.

Since the dual ofG1,h−1 isGh−1,1, the Serre dual of a πL-divisible OL-module can

be used to define a map

∨: LT_h∞ → M_h∞.

However, this map is notWL-equivariant. For example, the Tate module of the Serre

dual of(L/O_L)h isOh

L(1), and notO h

L. As such, while the duality map∨could be

used to study the cohomology ofM_h∞as aE× ×GLh(L)-representation, the action

ofWL appears to be difficult to understand.

Instead, we will use the exterior power Vr

, which was introduced by Pink and studied by Hedayatzadeh in [Hed10], [Hed13] and [Hed14] for certainπ_L-divisible

O_L-modules. The idea of using the exterior power has previously been explored by Chen in [Che13] where she used the top exterior power to define a determinant map

from Rapoport-Zink spaces to a tower of dimension 0 rigid analytic spaces. In this

thesis, however, we will consider the h−1 exterior power, and not the top exterior power. AsVh−1

G1,h−1 ≈ Gh−1,1, using the h−1 exterior power, we can define a map

h−1 ^

:LT_h∞ → M_h∞. Unlike the duality map ∨, the map Vh−1

is WL-equivariant, and the other group

actions can be understood as well. Under the mapVh−1

, the action of φ ∈ D× on LT_h∞ corresponds to the action ofθ(φ) on M_h∞, and the action of g ∈ GLh(L) on

LT_h∞corresponds to the action ofψ(g)on M_h∞.

Before proving Theorem B, it is useful to first look at the level 0 case. Using

Grothendieck-Messing theory and the fact that at level 0, the duality map∨induces an isomorphism on the connected components, we can show that the same is true for

the mapVh−1

of heightm, and define(M_hn)m similarly. By the above, we can identify(LT_h0)0with

(M_h0)0using the exterior power mapVh−1.

For simplicity, we will first specialize to the case whereh−1 is coprime top(q−1). Under this condition, we can check that K = ker(θ) is trivial, the map θ gives a bijection from O×

D to O

×

E, and θ maps a uniformizer of D

× _{to an element of E}×

with normalized valuation h−1. The fact that θ gives a bijection from O× D toO

×

E

shows thatVh−1

induces a bijection from the connected components of (LT_hn)0 to the connected components of(M_hn)0. Using this, and the fact that(LT_hn)0and(M_hn)0 are both étale covers of (LT_h0)0 of the same degree, we deduce thatVh−1 induces an isomorphism from (LT_hn)0 to (M_hn)0. By considering the group actions, it is then clear that Vh−1 _{induces an isomorphism from}

(LT_hn)m to (M_hn)(h−1)m. With

the above knowledge of the geometry of Vh−1 _{: LT}∞

h → M

∞

h , we can then prove

Theorem B in this special case.

Now, let us look at the general case. Unlike in the previous case where h−1 is coprime top(q−1), the mapVh−1

no longer induces a bijection from the connected

components of (LT_hn)0to the connected components of (M_hn)0. However, it is still true that each connected component of LT_hn is mapped isomorphically onto some connected component of M_hn. Letd = (h−1,q−1) and e = vp(h−1). We will

separately consider the cases where

(i) d _,1 ande =0,

(ii) d =1 ande _,0.

It should then be clear how to handle the general case.

First, we recall that(LT_hn)0 hasqn−1(q−1) connected components and a result of Strauch ([Str06]) tells us that the action of φ ∈ O×_D on the connected components depends on det(φ)mod(1 + π_LnO_L). We would like to understand the map on the connected components induced by Vh−1_{. To do so, we study the map} θ

0,n :

(O_L/πn_LO_L)× → (O_L/πn_LO_L)× induced by θ : O×

D → O

×

E under the det mod(1+ πn

LOL) map. In fact,θ0,nis simply the map x7→ x h−1_.

In case (i), the kernelK of the mapθ: O× D → O

×

E is cyclic of orderdand this maps

isomorphically to ker(θ0,n)under the det mod(1+πn_LOL)map. Hence, if we write

Ynfor the image of the mapVh−1: LT_hn→ M_hn, then we have

ψ∗

H_ci(Yn,Ql) θ∗* ,

H_ci(LT_hn,Ql)

K +

-asθ(D×)×GLh(L)×WL representations.

Let Eθ be the subgroup of E× generated by Im(θ) and O×

Lh, and let Xn be the

subspace`

m∈Z(M n

h)(h−1)m of M n

h. It is easy to check that, in fact,E

θ _{= {ϕ ∈ E}× _:

(h−1)|val(ϕ)}. Using thatEθ/Im(θ)is a finite abelian group, and thatXnis equal

to the disjoint union

Xn =

a

ϕ∈Eθ/Im(θ)

ϕ(Yn),

we can show that H_ci(Xn,Ql) IndE

θ

θ(D×₎(H_ci(Y_n,Q_l)). Similarly, E×/Eθ is finite

abelian (in fact cyclic of orderd), and

M_hn= a

ϕ∈E×_/Eθ

ϕ(Xn),

so we have H_ci(M_hn,Ql) IndE × Eθ(H

i

c(Xn,Ql)). Putting these together gives us

Theorem B for case (i).

We now look at case (ii). In this case, the kernel of the map θ : D× → E× is K = µ_pe(L), the group of pe roots of unity in L. Unlike in the previous case, K

does not surject onto ker(θ0,n) under the det mod(1+π_LnOL) map. But if we let

Kn = 1+πn_L−eeLOL ≤ D×, then the det mod(1+ πn_LOL)map gives a surjection of

K×Knonto ker(θ0,n), and we can show that

ψ∗

H_ci(Yn,Ql) θ∗* ,

H_ci(LT_hn,Ql)

K×Kn

+

-.

Now, by following the argument in case (i), we have

ψ∗

ResGLh_ψ(GLh(L)_(L))H_ci(M_hn,Ql)

IndE

θ

θ(D×₎θ∗* ,

H_ci(LT_hn,Ql)

K ×Kn

+

-.

The map H_ci(LT_hn−eeL,_Ql) → H_ci(LT_hn,_Ql) factors through H

i

c(LThn,Ql)

Kn since Kn =

1+πn_L−eeLO_L acts trivially onLT_hn−eeL, so we have

lim

−−→ n

H_ci(LT_hn,Ql)

Kn =

H_ci(LT_h∞,Ql).

Piecing together the above, we obtain Theorem B for case (ii).

Using Theorem B, and applying Frobenius reciprocity, we see that the following

diagram commutes:

RepE×

ρ7→θ∗_ResE×

θ(D×)ρ

/

/

Hi c(Mh∞)

RepD×

Hi c(LTh∞)

Rep(GLh(L)×WL)

π⊗r7→ψ∗_ResG Lh(L)

ψ(_{G Lh}(L))π

⊗r

/

/_{Rep(GLh(L)×WL)}

Although the above diagram only gives usH_ci(M_h∞)[ρ] as aψ(GLh(L))-representation,

we can use the duality map to understand theGLh(L)-action. Alternatively, in the

special case wherep(q−1)is coprime toh−1, the groupGLh(L)is generated by the

subgroupψ(GLh(L))and the elementπL ∈GLh(L), so we just need to understand

the action ofπL, but the action ofπLIh ∈GLh(L)onM_h∞corresponds to the action

ofπ−_L1 ∈ E×.

Finally, we observe thatθ∗ResE_θ(×_D×₎ ρ

andψ∗Res_ψGLh_(GLh(L)_(L))π can be written as twists ofρandπrespectively by their central characters. This is useful since both the local Langlands and the Jacquet-Langlands correspondences are compatible with

twists. Using the theorem of Harris-Taylor on the cohomology of the Lubin-Tate

tower, we can then prove Theorem A.

We start by introducing some background material on πL-divisible OL-modules in

Chapter 2. In particular, we recall the Serre dual of aπL-divisibleOL-module and

introduce the exterior power for πL-divisible OL-modules of dimension 1, whose

existence has been proven in [Hed10].

In Chapter 3, we introduce moduli spaces of certainπL-divisibleOL-modules, which

includes, as special cases, the Lubin-Tate and the dual Lubin-Tate towers. We also

state the well known result of Harris-Taylor on the cohomology of the Lubin-Tate

tower.

We collect some useful results on the endomorphism algebras and on representation

theory in Chapter 4. We will describe in this chapter embeddings ofD×andE×into GLh(Lh), and study properties of the mapsθandψ.

In Chapter 5, we use the exterior power introduced in Chapter 2 to define a map

And finally, in Chapter 6, we study the geometry of the dual Lubin-Tate tower, and

Notation

p: odd prime

l: odd prime different from p L: finite extension ofQp πL: uniformizer of L

q: order ofkL, the residue field of L

nL: degree of LoverQp

eL: ramification degree ofLoverQp

k: perfect field of characteristicpcontainingkL =Fq

WL: Weil group ofL

WO_L(k): ramified Witt vectors ofk

D(G): DieudonnéO_L-module of theπL-divisibleOL-moduleG

Gm,n: isoclinicπL-divisibleOL-module overFpof dim mand heightm+n

Dm,n: endomorphism algebra ofGm,n

G∨: Serre dual of theπL-divisibleOL-moduleG

Vr_G:

r-th exterior power of theπ_L-divisibleO_L-moduleG D: endomorphism algebra ofG1,h−1

E: endomorphism algebra ofGh−1,1

LTh: Lubin-Tate space, moduli space of deformations ofG1,h−1with a quasi-isogeny LT_hn: étale cover over the generic fiber ofLThparameterizing levelnstructure

LT_h∞: Lubin-Tate tower, the projective system{LT_hn}_n∈N

Mh: dual Lubin-Tate space, moduli space of deformations of Gh−1,1 with a quasi-isogeny

M_hn: étale cover over the generic fiber ofMhparameterizing levelnstructure

M_h∞: dual Lubin-Tate tower, the projective system {M_hn}_n∈_N

C h a p t e r 2

BACKGROUND ON

π

-DIVISIBLE

O

-MODULES

2.1 Ramified Witt vectors, DieudonnéO_L-modules andO_L-displays

Letpbe an odd prime,Lbe a finite extension of_Qpwith uniformizerπ_L and residue field kL = Fq. For an OL algebra R, the ring of ramified Witt vectorsWO_L(R) (cf.

[Haz80]) isWO_L(R)= RN with the uniqueOL-algebra structure such that

1. the Witt polynomials

wn :WOL(R) → R

(x0,x1,x2, . . .) 7→ xq

n

0 +πLx

qn−1

1 +· · ·+π

n Lxn

areO_L algebra homomorphisms for alln≥ 0,

2. O_L-algebra homomorphisms R → S induce O_L-algebra homomorphisms WOL(R) →WOL(S).

There is an endomorphism ofWOL(R), defined bywn(σ(x)) = wn+1(x), called the

Frobenius endomorphism. In fact,σ is a homomorphism ofO_L-algebras.

Let k ⊇ kL be a perfect field of characteristic p. ThenWO_L(k) OLD⊗W(kL)W(k),

whereW(k) =W_Zp(k)is the usual ring of Witt vectors ofk. It is a complete discrete valuation ring with residue field k and uniformizer πL. In particular,WOL(Fp) is

isomorphic to the ring of integers O_L˘ of ˘L, where ˘L = L·QLnr_p is the completion of

the maximal unramified extension of L.

We now introduce Dieudonné O_L-modules and O_L-displays. A Dieudonné O_L -module over k is a free WOL(k)-module of finite rank, together with two maps F

and V such that F is σ-linear, and V is σ−1-linear, and FV = πL = V F. There

is an equivalence of categories between the category of πL-divisible formal OL

-modules over k and the category of Dieudonné O_L-modules over k such thatV is topologically nilpotent in theπL-adic topology (cf. [Ahs11], Proposition 2.2.3 and

Theorem 5.3.8).

For aπ_L-divisibleO_L-moduleG, we will denote its covariant DieudonnéO_L-module by D(G). Under this equivalence of categories,

ht(G)= rank_WO

L(k) D(G), dim(G) =dimk

D(G)

Generalizing earlier work of Zink ([Zin02]) and Lau ([Lau08]), Ahsendorf ([Ahs11])

introduced the notion of an O_L-display. An O_L-display over an O_L-algebra R is a quadruple P = (P,Q,F,V−1) with P a finitely generated projective WOL(R)

-module, Q ⊂ P a submodule and F : P → P andV−1 : Q → P Frobenius linear maps, which satisfy some additional conditions.

IfπLis nilpotent inR, then there is an equivalence of categories between the category

of nilpotentO_L-displays overRand the category of formalπ_L-divisibleO_L-modules overR(cf. [Ahs11], Theorem 5.3.8).

2.2 IsoclinicπL-divisibleOL-modules and their endomorphism algebras

We now introduce certain isoclinic πL-divisible O_L-modules which will be useful later. Suppose m, n are coprime non-negative integers. We shall write Gm,n for

theπL-divisibleOL-module overFpwith covariant DieudonnéOL-moduleD(Gm,n)

having basis{b0,b1, . . . ,bm+n−1}such that

F bi = bi+n, V bi = bi+m and b(m+n)+i = πLbi.

Gm,nis an isoclinicπL-divisibleOL-module of dimensionmand heightm+n.

In particular, D(G1,h−1)has a basis{e0, . . . ,eh−1}such that

V ei= ei+1 (0≤ i ≤ h−2), V eh−1= πLe0,

andD(Gh−1,1)has a basis{f0, . . . , fh−1}such that

F fi = fi−1 (1≤ i ≤ h−1), F f0= πLfh−1.

LetO_Dm,n =End(Gm,n)be the endomorphism ring ofGm,nand letDm,n= ODm,n⊗Zp

Qp = End(Gm,n) ⊗Zp Qp be the endomorphism algebra of Gm,n. The following

description of the endomorphism ring O_Dm,n is a straightforward generalization of [JO99, Lemma 5.4]. We include a proof here for lack of a suitable reference.

Lemma 2.2.1. Leta,b ∈_Zbe such thatam+bn =1, and let O_Lm+n =WOL(Fm+n) be the ring of integers of Lm+n, the unramified extension of degreem+n over L. Then

O_Dm,n =End(Gm,n) OLm₊n[πm,n],

where λ · πm,n = πm,n · σb−a(λ) for λ ∈ O_Lm+n, and σ is the Frobenius map. λ ∈ O_Lm+n acts on b0 via multiplication by λ, and πm,n is a uniformizer of the

Proof. Using the above basis{b0,b1, . . . ,bm+n−1}ofD(Gm,n), we easily check that

πm,n(bi) =bi+1 (0≤ i ≤ h−2), πm,n(bh−1) = πLb0,

is an element of End(Gm,n).

Suppose φ ∈ End(Gm,n) is such that φ(b0) = λb0 for some λ ∈ O_L˘. Note that

(FbVa)i(b0) = bi, so

φ(bi) = φ((FbVa)ib0) = (FbVa)iφ(b0) = (FbVa)i(λb0)= σ(b−a)i(λ)bi.

In particular,

λb0= φ(b0) = φ(π−_L1bm+n)= σ(b−a)(m+n)(λ)π−L1bm+n= σ(b−a)(m+n)(λ)b0,

which shows that σ(b−a)(m+n)(λ) = λ. If we let a0 = a+n and b0 = b−m, then a0m+b0n= 1, so similarly, we haveσ(b0−a0)(m+n)(λ) = λ. But (b−a,b0−a0) = 1 since(n+a)(b−a)−a(b0−a0) = am+bn=1, soσm+n(λ) = λandλ ∈ O_Lm+n. The fact that such an element does indeed lie in End(Gm,n) is again easy to check. We shall denote this element of End(Gm+n)byλ.

Now, let φ be an arbitrary element of End(Gm,n). Then φ(b0) = Pm+n−1

i=0 αibi for some α_i ∈ O_L˘. A similar argument to that above shows, in fact, that α_i ∈ O_Lm+n. Clearly, φ(b0) −Pm+n−1

i=0 αiπim,n(b0) = 0, from which it follows that φ = Pm+n−1

i=0 αiπim,n. This proves that End(Gm,n) OLm+n[πm,n].

To show thatO_Lm+n[πm,n] is a discrete valuation ring with uniformizerπm,n, we first note that O_Lm+n[πm,n] is a subring of the division ring Lm+n(πm,n). Now, define a

mapv: Lm+n(πm,n) →Z∪ {∞}by

α0+α1πm,n+· · ·+αm+n−1πmm,n+n−17→min(m+n)vLm+n(αi)+i: 0 ≤ i < m+n .

It is easy to check thatvis a valuation onLm+n(πm,n). Since

O_Lm+n[πm,n]= {φ∈ Lm+n(πm,n) :v(φ) ≥ 0},

andv(πm,n) =1, we have shown that End(Gm,n) = O_Lm+n[πm,n] is a discrete valuation ring with uniformizerπm,n.

Let us introduce some notation for the cases(m,n)= (1,h−1)and(m,n) = (h−1,1). We write

O_D =O_D1,h−1 =End(G1,h−1), D = D1,h−1= End(G1,h−1) ⊗ZpQp, O_E = O_Dh−1,1 = End(Gh−1,1), E = Dh−1,1 =End(Gh−1,1)⊗Zp Qp.

Also let$(respectively$0) be the uniformizer ofO_D(respectivelyO_E) as in Lemma 2.2.1.

2.3 Serre dual and exterior powers ofπL-divisibleOL-modules

Let G be a π_L-divisible O_L-module over a scheme S. The dual π_L-divisible O_L -moduleG∨is

0→ (G[πL])∨→ (G[π_L2])∨→ (G[π3_L])∨→ · · ·

where(G[πi_L])∨=HomS(G[πi_L],Gm)is the Cartier dual ofG[πi_L]. We have

ht(G) =dim(G)+dim(G∨) =ht(G∨).

If S = Speck for some perfect fieldk of characteristic p, thenG∨ has Dieudonné

O_L-module

D(G∨) D(G)∨= Hom_WO

L(k)(D(G),WOL(k))

with(Fl)(v)= σ(l(Vv)),(V l)(v)= σ−1(l(Fv)), forv ∈ D(G),l ∈D(G)∨. We will now look at the exterior power ofπL-divisibleOL-modules, which has been

shown to exist by Hedayatzadeh in [Hed10] in certain situations. For G and H

πL-divisibleOL-modules over a base scheme S, we will write Alt

O_L

S (G

r_{,H)for the}

group of alternatingO_L-multilinear morphisms fromGr toH.

Definition 2.3.1. [Hed10, Definition 5.3.3] LetG,G0beπL-divisible OL-modules

over a scheme S, with an alternating O_L-multilinear morphism λ : Gr → G0such that for allπ_L-divisibleO_L-modulesH overS, the induced morphism

λ∗

: HomS(G0,H)→ AltO_SL(Gr,H) ψ7→ ψ◦λ

Theorem 2.3.2. [Hed10, Theorem 9.2.36] LetSbe a locally NoetherianO_L-scheme andGbe aπL-divisibleOL-module of heighthand dimension at most1. Then there exists a πL-divisible OL-module VrG over S of height

_h

r

, and an alternating morphism λ : Gr → Vr_{G such that for every morphism f : S}0 _→

S and every πL-divisibleOL-moduleH overS0, the homomorphism

HomOL S0 (f

∗

r

^

G,H) →AltOL S0 ((f

∗

G)r,H)

induced by f∗λis an isomorphism. In other words,Vr

Gis ther-th exterior power ofG over S, and taking the exterior power commutes with arbitrary base change. Moreover, the dimension ofVr

Gis a locally constant function

dim : S→

(

0, h−1 r−1

! ) s7→       

0 ifGsis étale,

_h−1

r−1

otherwise.

In the cases that Sis the spectrum of a perfect field of char p > 2, or the spectrum of a local ArtinO_L-algebra with residue charp> 2, we can write down the exterior power in terms of its DieudonnéO_L-module or itsO_L-display.

Theorem 2.3.3. [Hed10, Construction 6.3.1, Lemma 6.3.2, Proposition 6.3.3,

Sec-tion 9.1] LetP = (P,Q,F,V−1) be anO_L-display of height hwith tangent module of rank one. Fix a normal decomposition

P= L⊕T, Q= L ⊕IRT.

Then

r

^

P= *

, r ^ P, r ^ Q,

r−1 ^

V−1∧F,

r

^ V−1+

-is anO_L-display of heighth_r and rank h_r−−₁1. The map

λ: Pr → r

^ P

(x1, . . . ,xr) 7→ x1∧ · · · ∧xr

is an alternating morphism ofO_L-displays and satisfies the universal property that the map

Hom( r

^

P,P0)→ Alt(Pr,P0)

Theorem 2.3.4. [Hed10, Proposition 9.2.24] LetRbe a local ArtinO_L-algebra, and

GaπL-divisibleOL-module overRwith special fiber that is a connectedπL-divisible O_L-module of dim 1. Let P be the O_L-display of the πL-divisible OL-module G. Then theO_L-display ofVr_GisVr_P.

As an immediate corollary, forG aπ_L-divisibleO_L-module of dimension 1 over a perfect field of characteristicp, the DieudonnéO_L-module ofVr_{Gis just}Vr _D(G),

whereFandV are as in Theorem 2.3.3.

So using the above description ofD(G1,h−1), we see that the DieudonnéOL-module

ofVh−1

G1,h−1has a basis ˜e0, . . . ,e˜h−1, where

˜

ei = (−1)ie0∧e1∧ · · · ∧ei−1∧ei+1∧ · · · ∧eh−1,

and

Fe˜i = (−1)h−1e˜i−1 (1≤ i ≤ h−1), Fe0˜ = (−1)h−1πLe˜h−1, Ve˜i = (−1)h−1πLe˜i+1 (0≤ i ≤ h−2), Ve˜h−1= (−1)h−1e˜0.

C h a p t e r 3

MODULI SPACES OF

π

-DIVISIBLE

O

-MODULES

3.1 Basic Rapoport-Zink spaces

LetCbe the category of schemesSover SpecO_L˘ such thatpis locally nilpotent on S. Let ¯Sbe the closed subscheme ofSdefined by the ideal sheafpO_S.

We consider a special case of the moduli spaces constructed by Rapoport-Zink in

[RZ96].

Theorem 3.1.1. [RZ96, Theorem 3.25 and Proposition 3.79] Leth,d ∈_Z+coprime, andGbe an isoclinicπL-divisibleOL-module overFpof heighthand dimd. LetF be the functor

C−→Set

S 7−→   



(X, β) :X is a connectedπL-divisibleOL-module/S

and β :G_S¯ → X_S¯ a quasi-isogeny

   



,

∼.

ThenF is representable by a formal schemeMd

h, which is formally locally of finite type overSpfO_L˘. Ford =1, the formal schemeM1

h is (non-canonically) isomorphic to`

i∈_ZSpfO_L˘[[t1, . . . ,th−1]].

Let M0_d h

be the rigid analytic generic fiber ofMd

h. We consider a special case of

the tower of rigid-analytic coverings ofM0_d h

introduced in [RZ96, 5.34]. We write

G for the universalπ_L-divisibleO_L-module onMd

h, andTπL

Gfor its Tate module. LetΓnbe the subgroup ofGLh(OL)consisting of matrices congruent toIh modπn_L.

Then we have a finite étale coverMn_d h

ofM0_d h

that parameterizes trivializations

αn:OhL

−→TπLG mod Γn.

DefineM∞_d h

to be the projective system(Mn_d h

)n∈N with maps

fmn:Mnd h

→ Mm_d h

(m≤ n)

Recall from Section 2.2 thatGd,h−dis an isoclinicπL-divisibleOL-module of height

hand dimd. Any twoπL-divisibleOL-modules with the same Newton polygon are

isogenous, so in the definition ofMn_d

h

above, we might as well takeG= Gd,h−d.

The groups Dd,h−d = End(Gd,h−d) ⊗ZpQp andGLh(L) act onM

∞

d h

. Let C be the

completion of an algebraic closure of ˘L. We also have an action of the Weil group WL onM∞d

h

after base change toC. We describe the actions here.

1. Action ofDd,h−d: γ ∈ Dd,h−d acts onMnd h

by

(X, β, αn)7→ (X, β◦γ−1, αn).

2. Action ofGLh(L): Any element g0 ∈ GLh(L) can be written as g0 = πk_Lg

with k ∈ _Zand g ∈ GLh(L)∩Math(OL). Hence it is sufficient to describe

the action of such elements.

So letg ∈GLh(L)∩Math(OL) andS → Md

h be a formal scheme. We have

an isomorphismαrig: (L/OL)h

−→ Xrig. Let ker(g) be the kernel of the map g : (L/OL)h → (L/OL)h, andYrigbe theπL-divisibleOL-module X

rig

αrig_(ker(g)) overSrig. In other words,

ker(g) αrig

//

 _

αrig_(ker(g))  _

(L/OL)h

αrig

/

/

g

Xrig

qrig

(L/OL)h αrig

/

/_Yrig ₌ Xrig

αrig_(ker(g))

Thengacts onM∞_d h

by

(X, β, α)7→ (Y,q◦ β, α)

where

• Y is aπL-divisibleOL-module over some admissible blow-up ofSwith

rigid analytic generic fiberYrig,

• qis the special fiber of the quotient mapX →Y which has generic fiber Xrig → _αrigXrig

(ker(g)) =Y

• α : Oh L

−→ TπLY has generic fiber αrig : Oh L

−→ TπLYrig which corre-sponds toαrig_{: (L}/O

L)h

−→Yrig.

3. Action of WL: Suppose w ∈ WL 7→ σn ∈ Gal(kL/kL) = Gal(Fp/Fq) ⊆

Gal(Fp/Fp)under the reduction map, whereσ = {x 7→ xp} ∈ Gal(Fp/Fp) is

the arithmetic Frobenius. Then the action ofwonM∞_d h

×_L˘ C is

w :(X, β, α) 7→ (Xw, βw◦F_G/n

Fp, α

w₎

where

• Xw = X×_S Sw, G(pn) = X ×

Fp,σnFp,

• βw :G(_¯pn)

S → X

w

¯

S is induced by β :GS¯ → XS¯,

• αw is the map

O_Lh= (O_Lh)w α w −−−−−−−→

TπLXw.

Let us introduce some notation for the casesd = 1 and d = h−1. Ford = 1, we will writeLT_hnforMn₁

h

andLT_h∞for the Lubin-Tate towerM∞₁ h

.

And ford= h−1, we writeM_hnforMn_h−1 h

,M_h∞for the dual Lubin-Tate towerM∞_h−1 h

.

3.2 Results on the Lubin-Tate tower

A well known result of Harris-Taylor tells us that thel-adic cohomology (cf. [Ber93]) ofLT_h∞×_L˘Crealizes the local Langlands and the Jacquet-Langlands correspondence. To be precise, let us consider the functor

H_ci(LT_h∞) : RepD× →Rep(GLh(L)×WL) ρ7→HomD×(Hi

c(LT

∞

h ×L˘ C,Ql), ρ),

and let [Hc(LT_h∞)(ρ)] denote the virtual representation(−1)h−1P_ih₌−₀1(−1)i[Hci(LTh∞)(ρ)].

Theorem 3.2.1. [HT01, Theorem VII.1.3] Let π be an irreducible supercuspidal representation ofGLh(L). Then

[Hc(LT_h∞)(JL−1(π))]=[π⊗rec(π∨⊗ (| · | ◦det) h−1

2 )(h−1)]

in the Grothendieck group, where r ec is the local Langlands correspondence and

Theorem 3.2.2. [Fal02, Section 6] Fori_, h−1,

H_ci(LT_h∞)cusp(ρ) =0

for any ρ∈RepD×, where H_ci(LT_h∞)cusp is the supercuspidal part ofHci(LTh∞).

An immediate corollary of Theorems 3.2.1 and 3.2.2 is:

Corollary 3.2.3. Letπ be an irreducible supercuspidal representation ofGLh(L). Then

H_ch−1(LT_h∞)cusp(JL−1(π)) = π⊗rec(π∨⊗(| · | ◦det)

h−1

2 _)(h−1).

Let (LT_hn)m be the subspace of LT_hn corresponding to quasi-isogenies of height

m. The following result of Strauch describes the action of O×

D ×GLh(OL) on the

connected components of(LT_hn)0×_L˘ C.

Theorem 3.2.4. [Str06, Theorem 4.4(i)] There exists a bijection

π0

(LT_hn)0×_L˘ C

→ (OL/πn_LOL)×

which is O×

D × GLh(OL)-equivariant if we let (φ,g) ∈ O

×

C h a p t e r 4

THE ENDOMORPHISM ALGEBRAS,

GL

AND

REPRESENTATION THEORY

4.1 The endomorphism algebras andGLh

Recall from Lemma 2.2.1 that

O_Dm,n =End(Gm,n) OLm₊n[πm,n],

whereπm,nis a uniformizer of the discrete valuation ringODm,n.

Using the basis{e0,e1, . . . ,eh−1}ofD(G1,h−1)introduced in Section 2.2, the proof of Lemma 2.2.1 shows that we get the following embedding of D× Lh($)\{0}

intoGLh(Lh):

ιD : D× ,→ GLh(Lh)

λ7→

* . . . . . . .

,

λ

σ−1_(λ)

. . .

σ−(h−1)_(λ) + / / / / / / /

-$7→

* . . . . . . . . . .

,

0 0 . . . 0 πL

1 0 . . . 0 0 0 1 . . . 0 0

... ... ... ... ...

0 0 . . . 1 0

+ / / / / / / / / / /

-.

embedding

ιE : E× ,→GLh(Lh)

λ 7→

* . . . . . . .

,

λ

σ−1_(λ)

. . .

σ−(h−1)_(λ) + / / / / / / /

-$0_7→

* . . . . . . . . . .

,

0 1 0 . . . 0 0 0 1 . . . 0

... ... ... ... ...

0 0 0 . . . 1

πL 0 0 . . . 0

+ / / / / / / / / / /

-ofE× Lh($0)\{0}intoGLh(Lh).

Lemma 4.1.1. (a) For anyφ∈ D×,

Nrd(φ) =det(ιD(φ))

where Nrd is the reduced norm of the central division algebra D over L. In particular, we havedet(ιD(φ)) ∈ L×. Furthermore,

val(φ) =vL(det(ιD(φ)))

wherevalis the normalized valuation on the division algebra D, andvL is the usual normalizedπ_L-adic valuation on L.

(b) The result in (a) holds withDreplaced byE, andιD replaced byιE.

Proof. We shall prove (a). The proof of (b) is almost identical. The map

D× Lh→ Mh(Lh) (φ, µ)7→ µ·ι_D(φ)

isL-bilinear, so we have an L-linear map

f : D⊗_L Lh → Mh(Lh) φ⊗ µ7→ µ·ι_D(φ).

Let us show that f is injective. Suppose f(φ⊗ µ) = Ih. Then µ· ιD(φ) = Ih, so ιD(φ) = µ−1Ih. But the only scalar matrices in Im(ιD) are of the form µIh where µ∈ L. Hence,φ⊗ µ= µ−1⊗ µ= 1, proving injectivity.

Since the dimensions of D⊗ Lh and Mh(Lh) overLh are both equal to h2, f is an

isomorphism. Hence

Nrd(φ) =det(f(φ⊗1))= det(ιD(φ)),

proving the first assertion.

The second part follows immediately since val(φ) = vL(Nrd(φ)).

We shall view D× and E× as subgroups ofGLh(Lh) using the embeddingsιD and ιE. Define the maps

θ :D× → E× ψ :GLh(L)→ GLh(L)

φ7→ (detφ)(φ−1)T, g7→ (detg)(g−1)T.

Lemma 4.1.2. Letd = (h−1,q−1)ande= vp(h−1). The map θ0 :O×_D → O_E×

φ7→ (detφ)(φ−1)T

(a) has kernel

ker(θ0) = µh−1(OL) = µpc_d(O_L) = {ζ ∈ O_L : ζp c_d

=1}

of orderpcd, wherecis the maximum integer≤ esuch thatO_L contains thepc

roots of unity. IfLis unramified overQp, thenc= 0andker(θ0) = µd(OL)has orderd.

(b) image

Im(θ0) = {ϕ ∈ O×_E : det(ϕ) ∈(O_L×)h−1}

which is a normal subgroup ofO× E.

Proof.

(a) Letφ∈ker(θ0). Since

detφ must be a (h−1) root of unity in O_L. By Hensel’s lemma, any root of xh−pe1 ₋₁ = _{0 in}

Fq lifts uniquely to a root in OL. SinceF×q is cyclic of order

q−1, and(h−1,q−1) = d, the mapx 7→ xh−pe d1 _{is an automorphism ofF}× q, so

µh−1

pe (Fp) = µd(Fp), which shows that µh−1(OL) = µp

e_d(O_L)= µ_pc_d(O_L). So φ∈ker(θ0) ⇒detφ ∈ µpc_d(O_L)⇒ (φ−1)T ∈ µ_pc_d(O_L) ⇒φ ∈ µ_pc_d(O_L).

IfLis unramified overQp, thenLdoes not contain any non-trivialp-th roots of

unity, soc= 0 andµ_h−1(OL)= µd(OL).

(b) Since det(θ0(φ)) = (detφ)h−1, and detφ ∈ O_L×, the inclusion

Im(θ0) ⊆ {ϕ ∈ O_E× : det(ϕ) ∈ (O×_L)h−1}

is clear. To show the reverse inclusion, we need to show that any ϕsatisfying det(ϕ) ∈ (O_L×)h−1lies in Im(θ0). Letγ ∈ O×_L be a(h−1)root of det(ϕ). Then

θ(γ(ϕT₎−1

) =det(γ(ϕT)−1)γ−1ϕ= γh−1(detϕ)−1ϕ= ϕ.

This proves the reverse inclusion.

Proposition 4.1.3. Letc,d,ebe as in Lemma 4.1.2. The map

θ: D× → E×

φ7→ (detφ)(φ−1)T

has

(a) kernel

ker(θ) = µ_pc_d(O_L)= {ζ ∈ O_L :ζp c_d

= 1}

of orderpcd, and

(b) image

Im(θ) = {ϕ∈ E× : (h−1)|val(ϕ), det(ϕ) ∈ (L×)h−1}.

Let Eθ be the subgroup of E× generated by the subgroups O×

Lh and Im(θ), i.e.

Eθ =hO×

Lh,Im(θ)i. Then

(d) the cokernel

coker(θ)= E

×

Im(θ) Eθ Im(θ) ×

E× Eθ

with

Eθ Im(θ)

O_E×

Im(θ0)

and E

×

Eθ Z

(h−1)_Z.

Proof. Parts (a) and (b) follow almost immediately from Lemma 4.1.2.

(c) It is clear thatEθ ⊆ {ϕ ∈E× : (h−1)|val(ϕ)}since

det(θ(φ)) = (detφ)h−1

andO×_Lh ⊆ O×_E.

Let us prove the reverse inclusion. It suffices to show that O× E ⊆ E

θ_{. An}

arbitrary element ofO×

E is of the form

ϕ = a0+a1$0+a2($0)2+· · ·+ah−1($0)h−1,

witha0∈ O×

Lhandai ∈ OLhfor alli.

We observe that the determinant map on D×, when restricted to the subgroup

O_L× h ≤ D

×

, is equal to the norm map NLh/L : O×_Lh → O_L×. Since Lh is an

unramified extension ofL, the map NLh/L :O_L×_h → O_L×is surjective.

Now, given any ϕ ∈ O×

E, since detϕ ∈ L

× _{by Lemma 4.1.1, we must have}

detϕ ∈ O×

L. By surjectivity of the map NLh/L : O

×

Lh → O

×

L, we can find

u ∈ O× Lh ≤ D

× _{such that det(u) =det(ϕ). Then}

θ(u(ϕT)−1) =det(u(ϕT)−1)u−1ϕ=u−1ϕ,

sou−1ϕ ∈ Eθand hence ϕ∈ Eθ. This proves the reverse inclusion.

(d) Define

E× Im(θ) →

Eθ Im(θ) ×

E× Eθ

ϕIm(θ) 7→ (ϕ$−bIm(θ), $bEθ)

wherebis any integer congruent to val(ϕ)mod(h−1). Since$h−1∈Im(θ) ⊆

Eθ, it is clear that this is well defined and is an homomorphism. By (iii), Eθ = {ϕ ∈ E× :(h−1)|val(ϕ)}, so the valuation map gives an isomorphism

E× Eθ

Z

from which surjectivity of the above map follows easily. For injectivity, we note

that

$b_∈

Eθ = {ϕ ∈E× : (h−1)|val(ϕ)} ⇒(h−1)|b⇒$b∈Im(θ). Hence

ϕ$−b_{∈Im(θ)and$}b _∈Eθ _⇒ϕ$−b _{∈Im(θ)and$}b _∈Im(θ) ⇒ϕ ∈Im(θ).

Now we consider the map

Eθ O

×

E

Im(θ0)

ϕ 7→ ϕ$−val(ϕ)Im(θ0)

Then

ϕ ∈Eθ lies in the kernel

⇔ϕ$−val(ϕ) ∈Im(θ0)

⇔ det(ϕ$−val(ϕ)) ∈(O_L×)h−1 (since(h−1)|val(ϕ) ⇒det($−val(ϕ)) ∈ (L×)h−1)

⇔ det(ϕ) ∈(L×)h−1 ⇔ϕ ∈Im(θ), so

Eθ Im(θ)

O×_E

Im(θ0)

.

Proposition 4.1.4. Letc,d,ebe as in Lemma 4.1.2, and let eL be the ramification degree of L over Qp. θ0 : O_D× → O_E× induces a map θ0,n : (OL/πn_LOL)× → (OL/πn_LOL)× making the following diagram

O_D×

θ0

det _//

O_L×

/

/// OL πn

LOL

×

θ0,n

O×

E det //O

×

L ////

OL πn

LOL

×

commute. θ0,nis the map

θ0,n :(OL/πnLOL)× → (OL/πnLOL)×

x 7→ xh−1.

(a) Im(θ0)is the preimage ofIm(θ0,n)under thedet mod(1+πn_LOL)map,

(b) thedet mod(1+πn_LO_L)map induces an isomorphism

coker(θ0) =

O× E

Im(θ0)

−→ (OL/π n LOL)

×

Im(θ0,n) =

coker(θ0,n),

(c) ker(θ0,n) can be written as a direct product ker(θ0,n) = K_n,0₁ × K0_n,2, where K0_n,2 = (1+πn_L−eeLO_L)/(1+πn_LO_L), and thedet mod(1+πn_LO_L)map restricts to an isomorphism

ker(θ0)

det mod(1+πn_LOL) −−−−−−−−−−−−−−→

K0_n,1.

In particular, ife =0, then thedet mod(1+πn_LO_L)map restricts to an isomor-phism fromker(θ0)toker(θ0,n).

Proof. To show thatθ0 :O_D× → O_E× induces a map

θ0,n:

O_L/πn_LO_L× → O_L/π_LnO_L×

making the above diagram commute, we need to check that ifφ ∈ O×

D satisfies

detφ≡1 mod (1+πn_LO_L),

then

detθ0(φ)≡ 1 mod (1+π_LnOL).

But this is clear since det(θ0(φ)) = (detφ)h−1. The fact thatθ0,nis the(h−1)power map is also immediate from this.

Forn1 ≥ n2, consider the map

coker(θ0,n1)=

O_L/πn1

L OL

×

(OL/πn_L1OL)×

h−1

O_L/πn2

L OL

×

(OL/πn_L2OL)×

h−1 =coker(θ0,n2).

Supposea ∈ O×

L is a (h−1)power modπ n2

L . By Hensel’s lemma, forn2> 2eeL, a

is also a (h−1) power modπn1

L , so the above map is a bijection for all n1 ≥ n2 >

2eeL. So, for all n1,n2 > 2eeL, we have

coker(θ0,n1)

=

coker(θ0,n2)

, and hence

ker(θ0,n1)

=

ker(θ0,n2) .

Now, let us assume that n > 2eeL. Sincen > 2eeL ⇔ 2(n−eeL) > n, it is clear

that

We note that

ker(θ0,n

0)

=

ker(θ0,n)

for alln

0 _≥ n ⇒

ker(θ0,n0)

1+π_Ln0−eeLO_L

=

ker(θ0,n)

1+πn_L−eeLO_L

for alln0 ≥ n.

Hence for anya0+· · ·+an−eeL−1π_Ln−eeL−1 ∈ker(θ0,n), there exists a uniquean−eeL ∈ O_L/π_Lsuch thata0+· · ·+an−eeL−1π_Ln−eeL−1+an−eeLπ_Ln−eeL ∈ker(θ0,n+1). Therefore, the set of elements a0+· · ·+ an−eeL−1π_Ln−eeL−1 ∈ ker(θ0,n) are given precisely by truncations of elements in ker(θ0). LetK_n,0₁ =ker(θ0)/(1+πn_LOL) ≤ ker(θ0,n). We have shown that ker(θ0,n) = K_n,0₁K_n,0₂. AsK_n,0₁∩K_n,0₂= {1}, this is a direct product, proving (c).

To prove (b), we first show that

coker(θ0) det

−−→

O_L× (O×_L)h−1. Consider the commutative diagram

1 //K //Im(θ _0) det //

(O_L×)_{ _}h−1 //

1

1 //K //O× E

det _//

O×

L //1

with exact rows, whereK = {ϕ∈ O×

E : detϕ= 1}. By the snake lemma,

coker(θ0)=

O× E

Imθ0 det

−−→

O× L (O×_L)h−1.

We have an isomorphism

O×_L µ_q−1×µ_pa×_ZnL_p

wherenL =[L:Qp] andais such that the group ofp-power roots of unity inO_L×is

µ_pa. So

coker(θ0) O×

L (O_L×)h−1

µ_q−1

(µ_q−1)h−1 ×

µ_pa (µ_pa)h−1

× Zp (h−1)Zp

!nL

µ_q−1

(µ_q−1)d ×

µ_pa

(µ_pa)pe ×

Z pe

Z !nL

Clearly, the kernel of the map

O_L× mod(1+ πn

LOL) −−−−−−−−−−−→

O_L/πn LOL

×

(OL/πn_LOL)×

h−1 = coker(θ0,n)

contains(O×_L)h−1, hence is equal to(O_L×)h−1forn > 2eeL, since

O× L (O_L×)h−1

= penL+cd =

ker(θ0,n)

=

coker(θ0,n) .

Therefore

coker(θ0) det

−−→

(O_L×) (O×_L)h−1

mod(1+πn LOL) −−−−−−−−−−−→

coker(θ0,n).

(a) then follows immediately from this.

Results analogous to those in Lemma 4.1.2 and Propositions 4.1.3, 4.1.4 also hold

for the maps

ψ0:GLh(OL)→ GLh(OL) ψ :GLh(L) →GLh(L)

g7→ (detg)(g−1)T, g7→ (detg)(g−1)T.

4.2 Representation theory

We start this section with a useful result that expresses the cohomology of certain

rigid analytic spaces in terms of the cohomology of a subspace.

Proposition 4.2.1. Let X be a rigid analytic space with an action of the group G, and letY be a subspace of X on which the subgroup H ofGacts. Suppose

(i) H is normal inG,

(ii) G_H is a finite abelian group,

(iii) X is the disjoint union

X = a

g∈G_H

g(Y).

Then

H_ci(X,Ql) IndGH H i

Proof. We first prove the result in the case that G_H is cyclic of orderm, generated by g0 ∈G. In this case, (iii) becomes

X =Y tg0(Y)t · · · tg₀m−1(Y).

So we have

H_ci(X,Ql) m−1 M

j=0

H_ci(Y,Ql)

as_Ql-vector spaces. Letxk be the element of

Lm−1

j=0 H

i

c(Y,Ql)withk-th coordinate

equal to xand which is 0 everywhere else. We pick the above isomorphism so that

g0· xk = xk+1, 0 ≤ k < m−1.

Define

f : IndG_H H_ci(Y,Ql)→ Hci(X,Ql) m−1 M

j=0

H_ci(Y,Ql)

g_okx7→ xk.

We will show that this is an isomorphism of representations.

Letg ∈G. Then we can write

g= hg₀l withh∈ H, 0 ≤ l ≤ m−1.

So it suffices to show that f is compatible with both the actions ofh ∈H and ofg0.

For 0 ≤ k < m−1,

g0· (g₀kx) =g₀k+1x

f

7−→ xk+1 =g0· xk =g0· f(g₀kx)

and sinceg₀m ∈ H,

g0·(g₀m−1x)= g₀0(g₀mx)

f

7−→ (g₀m · x)0= g₀m · x0=g0·xm−1 =g0· f(g₀m−1x).

And forh∈ H, we havehg₀k =g₀kh0for someh0∈ H sinceH is normal inG, so

h·(g₀kx)= g₀k(h0·x) 7−→f (h0·x)k =g₀k·(h0·x)0 = (g₀kh0)·x0= (hg₀k)·x0= h·xk = h·f(g₀kx).

So f is an isomorphism ofG-representations.

Now, for the general case where G_H is finite abelian, we can write G_H as a direct product G_H = G1

H × G2

H × · · · × Gn

eachGj

H is finite cyclic. LetHj =G1· · ·Gj. Since G

H is abelian,gj1Gj2 =Gj2gj1 for allg_j1 ∈ Gj1, 1 ≤ j1,j2 ≤ n, so Hj is a subgroup ofG and Hj−1 is normal in Hj. Furthermore, we haveHn= Gand _Hj−Hj₁ = G1

···Gj G1···Gj−1

Gj

(G1···Gj−1)∩Gj = Gj

H. DefineXj

recursively by X0 =Y and

Xj =

a

gj∈_Hj−Hj

1

gj(Xj−1).

(iii) implies that the above is indeed a disjoint union for all 1 ≤ j ≤ n, and that X = Xn. Applying the previous case repeatedly shows that

H_ci(X,Ql) IndHn_Hn−1IndHn−_Hn−1₂· · ·IndH_H1₀ Hci(Y,Ql) IndGH H i

c(Y,Ql).

Next, we include a result that will be needed later in Section 6.3 to show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower realizes the local

Langlands and the Jacquet-Langlands correspondences. This result is useful as local

Langlands and Jacquet-Langlands both behave well with respect to twists.

Lemma 4.2.2. (a) Let (ρ,V) be an irreducible representation of E×, and let ι : D× → E× be the isomorphism

D× ι

//

 _

E×_{ _}

GLh(Lh) _φ

7→(φT)−1

/

/_GLh(Lh).

Thenθ∗ Res_θE₍×_D×₎ ρ

ι∗ρ⊗(χι∗_ρ◦Nrd)∨, where χι∗_ρis the central character ofι∗ρ, andNrd :D → Lis the reduced norm map of the division algebraD.

(b) Letπbe an irreducible representation ofGLh(L), and let jbe the isomorphism

j :GLh(L)

−→GLh(L)

g 7→ (gT)−1.

Then ψ∗Res_ψGLh_(GLh(L)_(L))π j∗π ⊗ (χj∗_π ◦ det)∨, where χ_j∗_π is the central character of j∗π.

First, note that, by Lemma 4.1.1, for our embedding of D× into GLh(Lh), the

determinant map onGLh(Lh), when restricted to D×, has image inL, and is in fact

the reduced norm map Nrd :D× → L. In other words, the diagram

D×_{ _} Nrd //

L_{ _}

GLh(Lh) _det // Lh

commutes.

Forφ ∈ D×,v ∈V,

((θ∗

Resρ)(φ))(v)= (ρ((detφ)(φT)−1))(v) = (ι∗ρ((detφ)−1φ))(v).

Letι∗ρ⊗(χι∗_ρ◦Nrd)∨act on the vector spaceV ⊗

Ql HomQl(Ql,Ql). Consider the

isomorphism

V →V ⊗

Ql HomQl(Ql,Ql)

αv 7→ v⊗ fα (where fα : c7→αc).

Forφ ∈ D×,

(((χι∗_ρ◦Nrd)∨(φ))(fα))(c)= f_α((χι∗_ρ◦Nrd)(φ−1)(c))

= fα(ι∗ρ(detφ−1)(c))

= α(ι∗_ρ(

detφ−1))(c),

so((χι∗_ρ◦Nrd)∨(φ))(f_α)= f_α

(ι∗_ρ(detφ−1₎₎, and

(ι∗_ρ(φ))(

v)⊗ ((χι∗_ρ◦Nrd)∨(φ))(f₁) = (ι∗ρ(φ))(v) ⊗ f_ι∗_ρ(detφ−1₎

7→ (ι∗ρ((detφ)−1φ))(v),

C h a p t e r 5

THE DUALITY MAP AND THE EXTERIOR POWER MAP ON

MODULI SPACES

5.1 Definitions of the duality map and the exterior power map

In this section, we will use the Serre dual and the exterior power of a πL-divisible O_L-module to define maps from LT_h∞ toM_h∞.

We start with the Serre dual. Note that G∨_m,n Gm,n, so G∨_1,h−1 Gh−1,1 has dimensionh−1 and heighth. Let us fix such an isomorphism. The dual of aπL

-divisibleO_L-moduleX has Tate module(TπLX)

∨

(1). In particular, the Tate module of the dual of (OL/πn_LOL)h is equal toO_Lh(1). Over ˘L(ζp∞), we haveOh

L(1) O h L

since ˘L(ζp∞)contains thep∞ roots of unity.

Definition 5.1.1. Define∨: LT_h∞×_L˘ L˘(ζp∞)→ M∞

h ×L˘ L˘(ζp∞)by (X, β, α) 7→ (X∨,(β∨)−1,(α∨_ζ

pn)

−1₎

whereα∨_ζ

pn is given by the composition

TπLX∨−−−−−−→ (TπLX)∨(1)−−−−−−→ O _Lh(1) −−−−−−→ O _Lh.

Let us now look at the exterior power. Recall thatVh−1

G1,h−1 Gh−1,1. We will fix such an isomorphism.

Definition 5.1.2. DefineVh−1

:LT_h∞ → M_h∞by

(X, β, α) 7→* ,

h−1 ^

X,

h−1 ^

β, h−1 ^

α+

-.

In the above, by a slight abuse of notation,Vh−1α_{is the level structure on}Vh−1 X given by

O_Lh −−−−−−→φ

h−1 ^

O_Lh

Vh−1α

−−−−−−→

h−1 ^

TpX −−−−−−→

Tp

*

,

h−1 ^

X+

-,

whereφis the isomorphism

v_i 7→ (−1)iv1∧v2∧ · · · ∧vi−1∧vi+1∧ · · · ∧vh,

5.2 Properties of the exterior power map and group actions

In order to study the cohomology of the dual Lubin-Tate tower using the exterior

power mapVh−1

, it is necessary to first understand how the group actions behave

with respect toVh−1 .

Recall from Section 4.1 that we have embeddings of D× and E× in GLh(Lh). We

shall view D× andE× as subgroups ofGLh(Lh) using these embeddings. We also

recall the maps

θ :D× → E× ψ :GLh(L)→ GLh(L)

φ7→ (detφ)(φ−1)T, g7→ (detg)(g−1)T.

Proposition 5.2.1. The mapVh−1 _:LT∞

h → M

∞

h is

(a) D×-equivariant if we letD× act on M_h∞ viaθ,

(b) GLh(L)-equivariant if we letGLh(L)act on M_h∞ viaψ,

(c) WL-equivariant.

Proof.

(a) Supposeφ ∈ D× is given by

φei = h−1 X

j=0

αjiej.

Then

*

,

h−1 ^

φ+

-˜

ei =(−1)iφe0∧φe1∧ · · · ∧φei−1∧φei+1∧ · · · ∧φeh−1

=(−1)i h−1 X

j=0

αj0ej∧ · · · ∧ h−1 X

j=0

αj,i−1ej ∧ h−1 X

j=0

αj,i+1ej∧ · · · ∧ h−1 X

j=0

αj,h−1ej

=

h−1 X

k=0

βk(e0∧e1∧ · · · ∧ek−1∧ek+1∧ · · · ∧eh−1)

for some βk ∈ Lh. To write down an expression for βk, it is helpful to first

introduce the order-preserving bijections

bl : {0,1, . . . ,l−1,l+1, . . . ,h−1} → {1,2, . . . ,h−1}

j 7→

     



Forτ ∈Sh−1, letτik =b−_k1◦τ◦bi. Then βk = (−1)i

X

τ∈Sh−1

sgn(τ)ατ_ik(0),0· · ·ατik(i−1),i−1ατik(i+1),i+1· · ·ατik(h−1),h−1.

So the coefficient of ˜ek in

_Vh−1

φ ˜

ei is given by

(−1)i+kdet(φwith row and column containing αk,ideleted)

=Ck,i, the cofactor of the elementak,iofφ.

So

h−1 ^ φ = * . . . . . . . ,

C0,0 C0,1 · · · C0,h−1 C1,0 C1,1 · · · C1,h−1

... ... . . . ...

Ch−1,0 Ch−1,1 · · · Ch−1,h−1

+ / / / / / / /

-= (detφ)(φ−1)T.

(b) It suffices to prove the proposition forg ∈GLh(L)∩Math(OL). For suchg, it

acts onLT_h∞ by

(X, β, α) 7→ (Y,q◦ β, α)

whereY,q, αare as defined in Section 3.1. Recall thatYrig,qrigandαrig_(which corresponds toαrig :O_Lh −→ TπLY

rig_{)satisfy the commutative diagram}

(L/OL)h α

rig // g Xrig qrig

(L/OL)h α

rig

/

/_Yrig ₌ Xrig

αrig_(ker(g)).

Applying the functorVh−1

to the above diagram gives

Vh−1

(L/O_L)h

Vh−1_αrig

//

Vh−1_g

Vh−1 Xrig

Vh−1_qrig

Vh−1₍

L/O_L)h

Vh−1αrig

//

Vh−1 Yrig.

By an argument similar to that in (a), the following diagram commutes:

O_Lh

ψ(g)=(detg)(g−1₎T

//Vh−1_Oh L

Vh−1_g

O_Lh

//

SinceVh−1_αrig_{corresponds to(}Vh−1_α)rig_{, using the above 2 diagrams, we get} the commutative diagram

Oh L

Vh−1α

//

ψ(g)

Tp

_Vh−1 X

Vh−1_q

Oh L

Vh−1_α

//Tp

_Vh−1 Y.

Henceψ(g)acts on M_h∞ by

*

,

h−1 ^

X,

h−1 ^

β, h−1 ^

α+

-7→*

,

h−1 ^

Y,

h−1 ^

q◦ h−1 ^

β, h−1 ^

α+

-,

as required.

(c) Letw ∈WL. By the universal property of the exterior power, we have

F₍Vh−1_G

1,h−1)/Fp =

h−1 ^

F_G

1,h−1/Fp,

and

Tp* ,

h−1 ^

X+

-=

h−1 ^

TpX.

The first equality shows that (Vh−1β

)w = Vh−1βw_{, and the second gives}

(Vh−1_α)w ₌Vh−1_αw_{, so the map}Vh−1_{: LT}∞

h → M

∞

h isWL-equivariant.

Proposition 5.2.1 tells us that the exterior power mapVh−1

isWL-equivariant. Unlike

Vh−1_{however, the duality map∨is not}

WL-equivariant, one of the reasons being that

the Tate module of the dual of(L/O_L)hisOh

L(1). As such, it is difficult to understand

theWL-action on M_h∞ by considering the duality map. However, the duality map∨

can still be used to study the cohomology of M_h∞as aE××GLh(L)-representation,

and hence can be used to show that the supercuspidal part of the cohomology ofM_h∞ in the middle degree realizes the Jacquet-Langlands correspondence up to certain

C h a p t e r 6

THE GEOMETRY AND COHOMOLOGY OF THE DUAL

LUBIN-TATE TOWER

6.1 Geometry of the dual Lubin-Tate tower

Before looking at the tower, it is helpful to first understand the level 0 situation. Let us writeLThfor the formal schemeM1

h, andMhforM h−1

h .

Proposition 6.1.1. The mapVh−1

:LTh → Mhinduces an isomorphism h−1

^

: (LTh)0→ (Mh)0.

Proof. By Theorem 3.1.1, we have a non-canonical isomorphism

(LTh)0≈SpfO_L˘[[t1, . . . ,th−1]],

which shows, by considering the duality map∨, that

(Mh)0≈SpfO_L˘[[T1, . . . ,Th−1]].

Let P = (P,Q,F,V−1) be the O_L-display corresponding to the πL-divisible OL

-module G1,h−1, so that P = D(G1,h−1) and Q = V D(G1,h−1) with F and V−1 as given in the definition of D(Gm,n). Let S ∈ C, and let SpecA ⊆ S be an open affine subset of S. By Grothendieck-Messing theory (cf. [Mes72]), πL-divisible O_L-modules over AliftingG1,h−1correspond to s.e.s.

0−→ M −→ P⊗O_L˘ A−→ N −→0

of A-modules lifting the Hodge filtration

0−→ Q

πLP −→ P

πLP −→ P

Q −→0. Since

Q

πLP =

V D(G1,h−1)

πLD(G1,h−1) =

hV e0, . . . ,V eh−1i

hπLe0, . . . , πLeh−1i =

hπ_Le0,e1, . . . ,eh−1i

hπLe0, . . . , πLeh−1i

,

the above condition is equivalent to

M mAM =

Q

πLP =

he1, . . . ,eh−1i

hπLe1, . . . , πLeh−1i

Applying Nakayama’s lemma, we see thatM must be of the form

he1+t1πLe0, . . . ,eh−1+th−1πLe0i

for somet1, . . . ,th−1∈ A. ApplyingVh−1

to the s.e.s.

0−→ he1+t1πLe0, . . . ,eh−1+th−1πLe0i −→ he0, . . . ,eh−1i −→ he0i −→0, (6.1)

we get the s.e.s.

0−→ h(e1+t1πLe0)∧· · ·∧(eh−1+th−1πLe0)i −→ he0˜, . . . ,e˜h−1i −→ he1˜, . . . ,e˜h−1i −→0

where

(e1+t1πLe0)∧ · · · ∧(eh−1+th−1πLe0)= e˜0−t1πLe˜1−t2πLe˜2− · · · −th−1πLe˜h−1.

Let us rewrite (6.1) as follows:

0−→ hf1, . . . , fh−1i

ρ

−→ he0, . . . ,eh−1i

τ −

→ hg0i −→0 fi 7→ ei+tiπLe0

e0 7→ g0

ei 7→ −tiπLgi, i ,0.

Applying∨, we get the s.e.s.

0−→ hg∨₀i τ ∨

−−→ he₀∨, . . . ,e∨_h−1i ρ ∨

−−→ hf₁∨, . . . , f_h∨₋₁i →0

where

(τ∨

g₀∨)(ei)= g∨0(τei)=

   

 



g₀∨(g0)= 1, ifi= 0,

g₀∨(−tiπLg0) =−tiπL, ifi, 0,

(ρ∨e_i∨)(fj)= e∨_i (ρfj)= e∨_i (ej +tjπLe0) =

     



1, ifi = j, 0, ifi _, j.

So the above s.e.s. is the same as

0−→ he∨₀−t1πLe∨₁− · · · −th−1πLe∨_h−1i −→ he

∨

0, . . . ,e

∨

h−1i −→ he

∨

1, . . . ,e

∨

Therefore we have a commuting diagram

(Mh)0≈ SpfO_L˘[[T1, . . . ,Th−1]]

(LTh)0≈SpfO_L˘[[t1, . . . ,th−1]]

Vh−1 22

∨

_,,

(Mh)0 ≈SpfO_L˘[[S1, . . . ,Sh−1]]

Ti_7→ Si

O

O

which shows that Vh−1 _{induces an isomorphism from (}

LTh)0 to (Mh)0 since the duality map∨: LTh→ Mhclearly does.

Corollary 6.1.2. The mapVh−1 _{: LT}

h → Mhinduces an isomorphism h−1

^

: (LTh)m → (Mh)(h−1)m

for anym∈_Z.

Proof. We recall the definition of the mapθ : D× → E×. Using the above embed-dings of D× andE×intoGLh(Lh), we haveθ(φ) = (detφ)(φ−1)T.

Letm∈_Z. Fix someφ ∈D×with val(φ) = m, then

val(θ(φ))= v_L(det(θ(φ))) = v_L det(detφ)(φ−1)T

= vL((detφ)h−1) = (h−1)vL(detφ)= (h−1)m.

So the action of φinduces an isomorphism from(LT_h0)0to (LT_h0)m, and the action

ofθ(φ) induces an isomorphism from(M_h0)0to(M_h0)(h−1)m.

By Proposition 5.2.1, we have the following commutative diagram:

(LT_h0)0

Vh−1

/

/

φ

(M_h0)0

θ(φ)

(LT_h0)m Vh−1 //(M 0

h)(h−1)m

where the top map is an isomorphism by Proposition 6.1.1. HenceVh−1

induces an

isomorphism from (LT_h0)m to (M_h0)(h−1)m, as desired.

Proposition 6.1.3. For anyn ≥ 0,m,n∈_Z, each connected component of (LT_hn)m is mapped isomorphically onto some connected component of(M_hn)(h−1)m under the mapVh−1